New Problems in Mean Field Control Theory

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Mean field games were introduced in 2006, independently by Lasry-Lions and Huang-Caines-Malhamé. Their idea was to adapt the mean field approach from physics and mechanics to social sciences. The mean field approach allows the description of the macroscopic behavior of fluids and gases without considering the motion of the individual particles. One can consider a representative particle whose dynamics is influenced by a mean field term that represents an averaging effect coming from all other particles. In the social sciences, particles are replaced with agents. For example, an agent can be a car when modeling traffic on highways. An important difference from physics is that the agents are decision makers and the dynamics correspond to decisions. This proposal develops control methods for these mean field models, called Mean Field Control Theory. Results will be applied to dynamic networks and provide insight for applications such as economic theory.A basic problem in mean field control theory is to solve a system of Hamilton-Jacobi-Bellman and Fokker Planck (HJB-FP) equations. The Master Equation, introduced by Lions, allows one to encapsulate the system of HJB-FP equations in a single equation by driving a decoupling argument. A serious difficulty is the fact that one needs to work with functions of probability measures. Lions introduced the idea that one could write the Master Equation using for the space of arguments the Hilbert space of square integrable random variables. This leads to new control problems in this particular Hilbert space, which is the first objective of the research. The second objective is to study specific problems in mean field control theory, motivated by economics.